The Recursion Method Applied to the T=0 Dynamics of the 1D s=1/2 Heisenberg and XY Models

The frequency‐dependent spin autocorrelation functions for the 1D s=1/2 Heisenberg and XY models at zero temperature are determined by the recursion method. These applications further demonstrate the efficacy of a new calculational scheme developed for the termination of continued fractions. A special feature of the recursion method highlighted here is its capability to predict the exponent of the infrared singularities in spectral densities

Dynamical Properties of Quantum Spin Systems in Magnetically Ordered Product Ground States

The one‐dimensional spin‐s XYZmodel in a magnetic field of particular strength has a ferro‐ or antiferromagnetically ordered product ground state. The recursion method is employed to determine T=0 dynamic structure factors for systems with s=1/2, 1, 3/2. The line shapes and peak positions differ significantly from the corresponding spin‐wave results, but their development for increasing values of s suggests a smooth extrapolation to the spin‐wave picture

Dynamics of Semi-infinite Quantam Spin Chains at T = ∞

Time-dependent spin autocorrelation functions and their spectral densities for the semi-infinite one-dimensiona ls=1/2 XY and XXZ models at T=∞ are determined in part by rigorous calculations in the fermion representation and in part by the recursion method in the spin representation. Boundary effects yield valuable new insight into the different dynamical processes which govern the transport of spin fluctuations in the two models. The results obtained for the XXX model bear the unmistakable signature of spin diffusion in the form of a square root infrared divergence in the spectral density

Dynamics of Semi-infinite Quantam Spin Chains at T = ∞

Time-dependent spin autocorrelation functions and their spectral densities for the semi-infinite one-dimensiona ls=1/2 XY and XXZ models at T=∞ are determined in part by rigorous calculations in the fermion representation and in part by the recursion method in the spin representation. Boundary effects yield valuable new insight into the different dynamical processes which govern the transport of spin fluctuations in the two models. The results obtained for the XXX model bear the unmistakable signature of spin diffusion in the form of a square root infrared divergence in the spectral density

Computation of Dominant Eigenvalues and Eigenvectors: A Comparative Study of Algorithms

We investigate two widely used recursive algorithms for the computation of eigenvectors with extreme eigenvalues of large symmetric matrices -- the modified Lanczös method and the conjugate-gradient method. The goal is to establish a connection between their underlying principles and to evaluate their performance in applications to Hamiltonian and transfer matrices of selected model systems of interest in condensed matter physics and statistical mechanics. The conjugate-gradient method is found to converge more rapidly for understandable reasons, while storage requirements are the same for both methods

Dimer and N\'eel order-parameter fluctuations in the spin-fluid phase of the s=1/2 spin chain with first and second neighbor couplings

The dynamical properties at T=0 of the one-dimensional (1D) s=1/2 nearest-neighbor (nn) XXZ model with an additional isotropic next-nearest-neighbor (nnn) coupling are investigated by means of the recursion method in combination with techniques of continued-fraction analysis. The focus is on the dynamic structure factors S_{zz}(q,\omega) and S_{DD}(q,\omega), which describe (for q=\pi) the fluctuations of the N\'eel and dimer order parameters, respectively. We calculate (via weak-coupling continued-fraction analysis) the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distribution in S_{zz}(\pi,\omega) and S_{DD}(\pi,\omega), all in the spin-fluid phase, which is realized for planar $nn$ anisotropy and sufficiently weak nnn coupling. For some parameter values we find a discrete branch of excitations above the spinon continuum. They contribute to S_{zz}(q,\omega) but not to S_{DD}(q,\omega).Comment: RevTex file (7 pages), 8 figures (uuencoded ps file) available from author

Zero-Temperature Dynamics of the One-Dimensional XXZ and t-J Models: A Weak-Coupling Continued-Fraction Analysis

We use the recursion method to study the spectral and dynamical properties of the one-dimensional (1D) s=1/2 XXZ method with planar anisotropy at T=0. Distinct methods of continued-fraction analysis have been developed for the weak-coupling and strong-coupling regimes of the corresponding lattice fermion system. The weak-coupling analysis presented here yields detailed information on the spectral-weight distribution in dynamic structure factors and spin autocorrelation functions, notably on the infrared singularities produced by critical fluctuations, and on the bound states for the case of attractive fermion interaction. The same method is then applied to the charge dynamics of the 1D t-J model for strongly correlated electrons. There it yields similar yet distinct results in the regime of weak exchange coupling. The results for renormalized bandwidths of particle-hole excitations are consistent with available results for charge velocities, and the results for the infrared exponent in the charge dynamic structure factor agree with existing results for the exponent of the equal-time charge correlation function

Ordering and fluctuations in the ground state of the one-dimensional and two-dimensional S = 1/2 XXZ antiferromagnets: A study of dynamical properties based on the recursion method

The recursion method is applied to the T = 0 dynamics of the S = 1/2 XXZ model on a linear chain and a square lattice. By means of new calculational techniques for the analysis of the continued-fraction coefficients pertaining to specific dynamical quantities, we obtain reliable information on the type of ordering in the ground state, on the size of gaps in the dynamically relevant excitation spectrum, on the bandwidths of dominant structures in spectral densities, on the exponents of infrared singularities, and on the detailed shape of spectral-weight distributions. We investigate some characteristic properties of the dynamic structure factors Sμμ(q, ω) and the spin autocorrelation functions Sμμ(ω) = N-1tsumqSμμ(q, ω), specifically their dependence on the uniaxial anisotropy, i.e., on the parameter which controls the type of ordering and the amount of quantum fluctuations in the ground state. We find, for example, that the different degrees of ordering in the planar regime of the one-dimensional and two-dimensional systems (criticality versus antiferromagnetic long-range order) have characteristic signatures in the dynamical properties which are conspicuously displayed in our results

Spin diffusion in the one-dimensional s = 1/2 XXZ model at infinite temperature

Time-dependent spin-autocorrelation functions at T = ∞ and (in particular) their spectral densities for the bulk spin and the boundary spin of the semi-infinite spin-1/2 XXZ model (with exchange parameters Jx = Jy = J, Jz) are investigated on the basis of (i) rigorous bounds in the time domain and (ii) a continued-fraction analysis in the frequency domain. We have found strong numerical evidence for spin diffusion in quantum spin models. For Jz/J increasing from zero, the results of the short-time expansion indicate a change of the bulk-spin xx-autocorrelation function from Gaussian decay to exponential decay. The continued-fraction analysis of the same dynamic quantity signals a change from exponential decay to power-law decay as Jz/J pproaches unity and back to a more rapid decay upon further increase of that parameter. By contrast, the change in symmetry at Jz/J = 1 has virtually no impact on the bulk-spin zz-autocorrelation function (as expected). Similar contrasting properties are observable in the boundary-spin autocorrelation functions